Electric Potential & Energy in Electric Fields Explained | Chapter 23 of University Physics

Chapter 23 introduces electric potential—a scalar, energy-based perspective on electrostatics that often simplifies calculations compared to vector force approaches. You’ll learn about electric potential energy, voltage (potential difference), equipotential surfaces, and how electric fields relate to potential through gradients and line integrals.

Watch the full video summary here for detailed derivations and examples.

Electric Potential Energy (U)

Electric potential energy (U) is stored due to charge configurations. The work done by the field equals the negative change in U:

U = (1 / 4πε₀) · (q · q₀) / r for two point charges, with U defined as zero at r → ∞. For multiple charges, sum over unique pairs:

U_total = Σ (1 / 4πε₀) · (q_i · q_j) / r_ij.

Electric Potential (V)

Electric potential (V) is the potential energy per unit test charge:

V = U / q₀, thus in volts (J/C). Potential difference between points a and b:

V_ab = V_a – V_b = W_a→b / q₀. For a point charge:

V = (1 / 4πε₀) · q / r. For many charges or continuous distributions, sum or integrate accordingly:

V = Σ (1 / 4πε₀) · q_i / r_i or V = (1 / 4πε₀) ∫ (dq / r).

Potential Difference & Electric Field

The relationship between E-field and potential is given by:

V_ab = – ∫_b^a E · dl, a path-independent line integral in electrostatics. Conversely, the field is the negative gradient of potential:

E = –∇V, with components E_x = –∂V/∂x, etc.

Example Geometries

• Conducting sphere: V = constant inside; outside behaves as V = (1 / 4πε₀)(Q/r).
• Parallel plates: Uniform E → V varies linearly across separation.
• Infinite line: V ∝ ln(r), found via ∫ E · dl.
• Ring or disk: Integrate dq/r to find V along the axis or point of interest.

Computing V can be simpler than directly finding the vector E in complex setups.

Equipotential Surfaces

Equipotential surfaces satisfy V = constant. Key properties:

• Electric field lines intersect equipotentials at right angles.
• No work is done moving a charge along an equipotential.
• Conductors in electrostatic equilibrium are equipotentials throughout.
• Inside a charge-free cavity, the potential is uniform (E = 0).

Conclusion

By mastering electric potential and its relationship to the field through gradients and integrals, you gain a powerful scalar method for analyzing electrostatic problems—often reducing multivariable vector challenges to simpler one-dimensional integrals.

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